Depends what you are talking about. Nearly all states communicate information. What the result of the communication is depends on the states.
In your particular situation, the info on your Texas license probably doesn't transfer to CA. When I moved between states, my driving record begins with a notation that I turned in a certain out-of-state drivers license to get my new one. If someone wants to pursue an in depth history on you, they'd use that to go query whatever info Texas had on you.
The other issue, and perhaps our Cali court clerk can answer is whether the old California conviction would be located on your new California driving record even without the assistance of the state of Texas. If I had to venture a guess, I would think not, but anything is possible in this day and age of computerized records.